The phrase "a proper class of" is correct and usable in written English.
It is typically used in mathematical or philosophical contexts to refer to a collection of objects that cannot be a member of any set due to size or other constraints.
Example: "In set theory, a proper class of all sets cannot be treated as a set itself."
Alternatives: "a valid category of" or "an appropriate group of".
where here AD+ is a strengthening of AD. (Strong Ω conjecture) Assume there is a proper class of Woodin cardinals.
However, if κ is a Woodin cardinal, then Vκ is a model of ZFC and Vκ satisfies that there is a proper class of strong cardinals.
Just as there can be no set of all sets, there can for diagonalization reasons also not be a proper class of all proper classes.
Every supercompact cardinal is Woodin, and if $\delta$ is Woodin, then $V_\delta$ is a model of ZFC in which there is a proper class of strong cardinals.
Summary: There is a "good theory" (one that freezes the theory of L and (granting a proper class of strongly inaccessible cardinals) all good theories imply ADL.
Thus, in contrast to CH, the Ω Conjecture cannot be shown to be independent of ZFC + "There is a proper class of Woodin cardinals" via set forcing.
Because S is a set (and not a proper class like the class of ordinal numbers), this construction ultimately must stop with a maximal member of S.
It is well-known that coding tricks allow one to do classical mathematics without ever going above cardinality c: for example, the class of all functions from the reals to the reals, is too large to be even a proper class here, but the class of continuous functions is of cardinality c.
The non-stationary ideal INS is a proper class from the point of view of H ω2) and it manifests (through Solovay's theorem on splitting stationary sets) a non-trivial application of AC.
Finally, the court ruled that appellant's suit was not a proper class action under Rule 23 of the Federal Rules of Civil Procedure.
There is a proper class von Neumann ordinal Ω, the class of all set von Neumann ordinals.
